Course

Unit 1

Learning Objectives
Understand the concept of phase space and its importance in Lagragian and Hamiltonian mechanics
Learn how to draw the phase portraits for different potentials and use it.
Applications of phase portrait in understanding the dynamics of physical systems.

Review of basic principles, Conservative systems, Conservation of linear momentum, Phase space-phase portrait – Dynamical Systems – Phase space dynamics – stability analysis.

Unit 2

Learning Objectives
Introduction of concepts of constraints, degrees of freedom
Limitations of Newtonian mechanics and the concept of generalized co-ordinate
Principal of virtual work, De- Alembert’s principle and Principle of Leas action
Lagrange’s Equation and its simple applications.

Lagrangian and Hamiltonian Mechanics with Constraints-Euler-Lagrange Equations, D’Alembert and Hamilton principles, Conservation Laws, holonomic and nonholonomic constraints – Generalized co-ordinates -Calculus of Variation, Principle of least action – The Lagrangian, Lagrange’s Equations, Degrees of Freedom, Generalized momentum & Hamilton’s Equations.

Unit 3

Learning objectives
Introduction to central forces
Bound states and scattering states
Concept of Lab frame and centre of Mass frame
Central forces – Kepler’s laws – bound state and scattering states. Determining the Motion using Energy Integral- Laboratory frame and centre of mass frame- Scattering.

Unit 4

Learning objectives
Concept of Rigid body
Introduce moment of Inertia tensor
Study rigid body rotation using Euler’s equation
Analysis of Symmetric top
Rotational Dynamics of Rigid Bodies: Conservation of Angular momentum, Moment of Inertia, Rotational Kinetic Energy, Euler Angles, Inertia Tensor, The Euler Equations-Analysis of a symmetric Top-Gyroscopes.

Unit 5

Learning objectives
Hamiltonian using Legendre transformation
Derivation of Hamilton’s equations
Apply Hamiltonian formulation to solve dynamical problems
Hamiltonian: Hamilton’s equations using Legendre Transformation- Cyclic co-ordinates- Application of Hamilton’s formalism in solving dynamical Problems.

Prerequisites: Mechanics, Mathematics 1

Course Objectives
This is the second course in mechanics and is intended to impart students basic understanding of other techniques used beyond Newtonian mechanics. Central forces, its applications in Kepler’s laws, Scattering will be discussed. An introduction to Rotational dynamics also will be covered.

Course Outcomes
At the end of the course, students will be able to

CO1: Introduction of Phase Space- Phase portrait and sketching the phase portraits of various potentials and its interpretation including applications.
CO2: Understand the concept of constraint, principle of least action and formulation of Lagrange’s method and apply Lagrange’s equation for simple dynamical systems.
CO3: Understand Central force and its application in Kepler’s problem and scattering problems. Use the Centre of mass and laboratory frames of references in solving problems.
CO 4: Understand the basics of rotating frames of references and Euler angles and Euler’s equations.
CO5: Apply Hamilton’s equations in solving dynamical problems.
Skill: Analytical skill to formulate dynamical problem and solve using Lagrangian Formalism.